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Plotter 6 Hyperboloid

Plotter 6 Hyperboloid
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2014-12-26 00:17
VectorFunctionCoordinate GeometrySolid GeometryParametric Equation
This example demonstrates the 3D geometric visualization and parametric construction of a hyperboloid. It involves core mathematical concepts such as coordinate geometry, solid geometry, parametric equations, and vector operations. Key GeoGebra commands and objects used in the construction include Corner, If, Mod, Element, Sequence, Polygon, Vector, Segment, Ray, Textfield, and ImplicitCurve. It is suitable for high school students and mathematics enthusiasts exploring complex spatial surfaces interactively.

GeoGebra Commands & Parameters

  1. Corner(1) → Eckpunkt1
  2. Corner(2) → Eckpunkt2
  3. If(c, α, (x(C - Eckpunkt1) / dEx * c_{Move})) → α1
  4. Mod(α1, 6.283) → α2
  5. Corner(3) → Eckpunkt3
  6. If(c, β, (y(C - Eckpunkt2) / dEy * c_{Move})) → β1
  7. Mod(β1, 6.283) → β2
  8. Element(lF, Funktion) → g0
  9. Sequence((a11' * Element[zWX1, i]) + (a12 * Element[zWY1, i]) + (a13 * Element[zWZ1, i]), i, 1, 4) → zWX1'
  10. Element(zWW_0, 3) → zW_2
  11. Element(zWW_0, 4) → zW_1
  12. Element(zWW_0, 2) → zW_3
  13. Polygon(Eckpunkt1, Eck11, Eck12, (x(Eckpunkt1), y(Eck12))) → Vieleck2, eckpunkt1, eck11, eck12, a_1
  14. Sequence(Segment[Element[zl10, i], Element[zl10, i + 1]], i, 1, 7) → BoxStr
  15. Vector(Element[zl10, 4], Element[zl10, 1]) → zVectx
  16. Vector(Element[zl10, 4], Element[zl10, 3]) → zVectzy
  17. Vector(Element[zl10, 4], Element[zl10, 5]) → zVectz
  18. Polygon(Eck11, Eckpunkt2, Eckpunkt3, Eck12) → Vieleck1, eck11_1, eckpunkt2, eckpunkt3, eck12_1
  19. Element(zTT, zW_1) → zY1
  20. Element(zTT, zW_2) → zY2
  21. Element(zTT, zW_3) → zY3
  22. Polygon(zTT) → zTTPoly, , , ,
  23. Element(zl10, zW_1) → zlN11
  24. Element(zl10, 9 - zW_1) → zlN111
  25. Element(zl10, zW_2) → zlN22
  26. Element(zl10, 9 - zW_2) → zlN222
  27. Element(zl10, zW_3) → zlN33
  28. Element(zWW_0, 1) → zW_4
  29. Element(zl10, zW_4) → zlN44
  30. Polygon({zlN11, zlN22, zlN222, zlN111}) → Vgrund01, , , ,
  31. Polygon(Sequence[Element[zl10, i], i, 1, 4]) → Vgrund000, , , ,
  32. Element(zl10, 7) → P_R
  33. Element(zl10, 8) → P_L
  34. Polygon({Element[zl10, 1], Element[zl10, 2], Element[zl10, 7], Element[zl10, 8]}) → Vvrd0, , , ,
  35. Segment(P_R, P_L) → StrRL
  36. Polygon(Sequence[Element[zl10, i], i, 9, 12]) → zlPoly, , , ,
  37. Ray(B, Element[zl10, 13]) → Stry
  38. Ray(B, Element[zl10, 14]) → Strzy
  39. Ray(B, Element[zl10, 15]) → Strx
  40. Textfield(x_{min}) → inputBox1
  41. Textfield(x_{max}) → inputBox4
  42. Textfield(v_{Animation}) → inputBox8
  43. Textfield(Step) → inputBox7
  44. Textfield(N) → inputBox9
  45. If(b_2, t, cos(((2 * π) * t0))) → Rot
  46. If(b_2, ℯ^(t), sin(((8 * π) * t0))) → Grün
  47. If(b_2, ℯ^((-t)), 1 - t0^(2)) → Blau
  48. Textfield(t) → inputBox10
  49. Textfield(g00) → textfield1
  50. Textfield(Funktion) → Textfeld1
  51. ImplicitCurve(wx3(x - x(B), y - y(B), t)) → ex1
  52. Vector(B, B2) → u
  53. Segment(B1, B) → e_2
  54. Polygon(Q1, Q3, Q4, Q2) → poly1, q1, q3, q4, q2
  55. Textfield(f) → inputBox2
  56. Textfield(Th) → inputBox3
  57. Textfield(g) → inputBox5
  58. Textfield(Stepg) → inputBox6
  59. Sequence(If[i > g, ¬j, j], i, 1, N + 1) → ltrue
  60. If(j, N + 1, 0) → trueb
  61. ImplicitCurve(wx3(x - x(B), y - y(B), (-4))) → M2_{1}
  62. ImplicitCurve(wx3(x - x(B), y - y(B), (-3.92))) → M2_{2}
  63. ImplicitCurve(wx3(x - x(B), y - y(B), (-3.84))) → M2_{3}
  64. ImplicitCurve(wx3(x - x(B), y - y(B), (-3.76))) → M2_{4}
  65. ImplicitCurve(wx3(x - x(B), y - y(B), (-3.68))) → M2_{5}
  66. ImplicitCurve(wx3(x - x(B), y - y(B), (-3.6))) → M2_{6}
  67. ImplicitCurve(wx3(x - x(B), y - y(B), (-3.52))) → M2_{7}
  68. ImplicitCurve(wx3(x - x(B), y - y(B), (-3.44))) → M2_{8}
  69. ImplicitCurve(wx3(x - x(B), y - y(B), (-3.36))) → M2_{9}
  70. ImplicitCurve(wx3(x - x(B), y - y(B), (-3.2800000000000002))) → M2_{10}
  71. ImplicitCurve(wx3(x - x(B), y - y(B), (-3.2))) → M2_{11}
  72. ImplicitCurve(wx3(x - x(B), y - y(B), (-3.12))) → M2_{12}
  73. ImplicitCurve(wx3(x - x(B), y - y(B), (-3.04))) → M2_{13}
  74. ImplicitCurve(wx3(x - x(B), y - y(B), (-2.96))) → M2_{14}
  75. ImplicitCurve(wx3(x - x(B), y - y(B), (-2.88))) → M2_{15}
  76. ImplicitCurve(wx3(x - x(B), y - y(B), (-2.8))) → M2_{16}
  77. ImplicitCurve(wx3(x - x(B), y - y(B), (-2.7199999999999998))) → M2_{17}
  78. ImplicitCurve(wx3(x - x(B), y - y(B), (-2.6399999999999997))) → M2_{18}
  79. ImplicitCurve(wx3(x - x(B), y - y(B), (-2.56))) → M2_{19}
  80. ImplicitCurve(wx3(x - x(B), y - y(B), (-2.48))) → M2_{20}
  81. ImplicitCurve(wx3(x - x(B), y - y(B), (-2.4))) → M2_{21}
  82. ImplicitCurve(wx3(x - x(B), y - y(B), (-2.3200000000000003))) → M2_{22}
  83. ImplicitCurve(wx3(x - x(B), y - y(B), (-2.24))) → M2_{23}
  84. ImplicitCurve(wx3(x - x(B), y - y(B), (-2.16))) → M2_{24}
  85. ImplicitCurve(wx3(x - x(B), y - y(B), (-2.08))) → M2_{25}
  86. ImplicitCurve(wx3(x - x(B), y - y(B), (-2))) → M2_{26}
  87. ImplicitCurve(wx3(x - x(B), y - y(B), (-1.92))) → M2_{27}
  88. ImplicitCurve(wx3(x - x(B), y - y(B), (-1.8399999999999999))) → M2_{28}
  89. ImplicitCurve(wx3(x - x(B), y - y(B), (-1.7599999999999998))) → M2_{29}
  90. ImplicitCurve(wx3(x - x(B), y - y(B), (-1.6800000000000002))) → M2_{30}
  91. ImplicitCurve(wx3(x - x(B), y - y(B), (-1.6))) → M2_{31}
  92. ImplicitCurve(wx3(x - x(B), y - y(B), (-1.52))) → M2_{32}
  93. ImplicitCurve(wx3(x - x(B), y - y(B), (-1.44))) → M2_{33}
  94. ImplicitCurve(wx3(x - x(B), y - y(B), (-1.3599999999999999))) → M2_{34}
  95. ImplicitCurve(wx3(x - x(B), y - y(B), (-1.2799999999999998))) → M2_{35}
  96. ImplicitCurve(wx3(x - x(B), y - y(B), (-1.1999999999999997))) → M2_{36}
  97. ImplicitCurve(wx3(x - x(B), y - y(B), (-1.12))) → M2_{37}
  98. ImplicitCurve(wx3(x - x(B), y - y(B), (-1.04))) → M2_{38}
  99. ImplicitCurve(wx3(x - x(B), y - y(B), (-0.96))) → M2_{39}
  100. ImplicitCurve(wx3(x - x(B), y - y(B), (-0.8799999999999999))) → M2_{40}
  101. ImplicitCurve(wx3(x - x(B), y - y(B), (-0.7999999999999998))) → M2_{41}
  102. ImplicitCurve(wx3(x - x(B), y - y(B), (-0.7199999999999998))) → M2_{42}
  103. ImplicitCurve(wx3(x - x(B), y - y(B), (-0.6400000000000001))) → M2_{43}
  104. ImplicitCurve(wx3(x - x(B), y - y(B), (-0.56))) → M2_{44}
  105. ImplicitCurve(wx3(x - x(B), y - y(B), (-0.48))) → M2_{45}
  106. ImplicitCurve(wx3(x - x(B), y - y(B), (-0.3999999999999999))) → M2_{46}
  107. ImplicitCurve(wx3(x - x(B), y - y(B), (-0.31999999999999984))) → M2_{47}
  108. ImplicitCurve(wx3(x - x(B), y - y(B), (-0.23999999999999977))) → M2_{48}
  109. ImplicitCurve(wx3(x - x(B), y - y(B), (-0.16000000000000014))) → M2_{49}
  110. ImplicitCurve(wx3(x - x(B), y - y(B), (-0.08000000000000007))) → M2_{50}
  111. ImplicitCurve(wx3(x - x(B), y - y(B), 0)) → M2_{51}
  112. ImplicitCurve(wx3(x - x(B), y - y(B), 0.08000000000000007)) → M2_{52}
  113. ImplicitCurve(wx3(x - x(B), y - y(B), 0.16000000000000014)) → M2_{53}
  114. ImplicitCurve(wx3(x - x(B), y - y(B), 0.2400000000000002)) → M2_{54}
  115. ImplicitCurve(wx3(x - x(B), y - y(B), 0.3200000000000003)) → M2_{55}
  116. ImplicitCurve(wx3(x - x(B), y - y(B), 0.40000000000000036)) → M2_{56}
  117. ImplicitCurve(wx3(x - x(B), y - y(B), 0.4800000000000004)) → M2_{57}
  118. ImplicitCurve(wx3(x - x(B), y - y(B), 0.5600000000000005)) → M2_{58}
  119. ImplicitCurve(wx3(x - x(B), y - y(B), 0.6399999999999997)) → M2_{59}
  120. ImplicitCurve(wx3(x - x(B), y - y(B), 0.7199999999999998)) → M2_{60}
  121. ImplicitCurve(wx3(x - x(B), y - y(B), 0.7999999999999998)) → M2_{61}
  122. ImplicitCurve(wx3(x - x(B), y - y(B), 0.8799999999999999)) → M2_{62}
  123. ImplicitCurve(wx3(x - x(B), y - y(B), 0.96)) → M2_{63}
  124. ImplicitCurve(wx3(x - x(B), y - y(B), 1.04)) → M2_{64}
  125. ImplicitCurve(wx3(x - x(B), y - y(B), 1.12)) → M2_{65}
  126. ImplicitCurve(wx3(x - x(B), y - y(B), 1.2000000000000002)) → M2_{66}
  127. ImplicitCurve(wx3(x - x(B), y - y(B), 1.2800000000000002)) → M2_{67}
  128. ImplicitCurve(wx3(x - x(B), y - y(B), 1.3600000000000003)) → M2_{68}
  129. ImplicitCurve(wx3(x - x(B), y - y(B), 1.4400000000000004)) → M2_{69}
  130. ImplicitCurve(wx3(x - x(B), y - y(B), 1.5200000000000005)) → M2_{70}
  131. ImplicitCurve(wx3(x - x(B), y - y(B), 1.6000000000000005)) → M2_{71}
  132. ImplicitCurve(wx3(x - x(B), y - y(B), 1.6799999999999997)) → M2_{72}
  133. ImplicitCurve(wx3(x - x(B), y - y(B), 1.7599999999999998)) → M2_{73}
  134. ImplicitCurve(wx3(x - x(B), y - y(B), 1.8399999999999999)) → M2_{74}
  135. ImplicitCurve(wx3(x - x(B), y - y(B), 1.92)) → M2_{75}
  136. ImplicitCurve(wx3(x - x(B), y - y(B), 2)) → M2_{76}
  137. ImplicitCurve(wx3(x - x(B), y - y(B), 2.08)) → M2_{77}
  138. ImplicitCurve(wx3(x - x(B), y - y(B), 2.16)) → M2_{78}
  139. ImplicitCurve(wx3(x - x(B), y - y(B), 2.24)) → M2_{79}
  140. ImplicitCurve(wx3(x - x(B), y - y(B), 2.3200000000000003)) → M2_{80}
  141. ImplicitCurve(wx3(x - x(B), y - y(B), 2.4000000000000004)) → M2_{81}
  142. ImplicitCurve(wx3(x - x(B), y - y(B), 2.4800000000000004)) → M2_{82}
  143. ImplicitCurve(wx3(x - x(B), y - y(B), 2.5600000000000005)) → M2_{83}
  144. ImplicitCurve(wx3(x - x(B), y - y(B), 2.6400000000000006)) → M2_{84}
  145. ImplicitCurve(wx3(x - x(B), y - y(B), 2.7199999999999998)) → M2_{85}
  146. ImplicitCurve(wx3(x - x(B), y - y(B), 2.8)) → M2_{86}
  147. ImplicitCurve(wx3(x - x(B), y - y(B), 2.88)) → M2_{87}
  148. ImplicitCurve(wx3(x - x(B), y - y(B), 2.96)) → M2_{88}
  149. ImplicitCurve(wx3(x - x(B), y - y(B), 3.04)) → M2_{89}
  150. ImplicitCurve(wx3(x - x(B), y - y(B), 3.12)) → M2_{90}
  151. ImplicitCurve(wx3(x - x(B), y - y(B), 3.2)) → M2_{91}
  152. ImplicitCurve(wx3(x - x(B), y - y(B), 3.2800000000000002)) → M2_{92}
  153. ImplicitCurve(wx3(x - x(B), y - y(B), 3.3600000000000003)) → M2_{93}
  154. ImplicitCurve(wx3(x - x(B), y - y(B), 3.4400000000000004)) → M2_{94}
  155. ImplicitCurve(wx3(x - x(B), y - y(B), 3.5200000000000005)) → M2_{95}
  156. ImplicitCurve(wx3(x - x(B), y - y(B), 3.6000000000000005)) → M2_{96}
  157. ImplicitCurve(wx3(x - x(B), y - y(B), 3.6799999999999997)) → M2_{97}
  158. ImplicitCurve(wx3(x - x(B), y - y(B), 3.76)) → M2_{98}
  159. ImplicitCurve(wx3(x - x(B), y - y(B), 3.84)) → M2_{99}
  160. ImplicitCurve(wx3(x - x(B), y - y(B), 3.92)) → M2_{100}
  161. ImplicitCurve(wx3(x - x(B), y - y(B), 4)) → M2_{101}

AI Drawing Prompt Reference

Visualize a hyperboloid by constructing its 3D geometric shape and parametric equations. Start with a basic grid formed by points, vectors, polygons, and rays to create a cubic framework. Then, use parametric equations to generate the surface. Key elements include vertices, vectors, polygons, and rays that define the structure and spatial relationships of the surface. The construction progresses from the base grid to the surface elements, resulting in a clear hyperboloid representation.
The prompt above was compiled from the original drawing commands and is for reference only.